\(A=\left(\frac{1}{4}-1\right)\left(\frac{1}{9}-1\right)\left(\frac{1}{16}-1\right)....\left(\frac{1}{400}-1\right)\)
\(=\frac{3}{4}\cdot\frac{8}{9}\cdot\frac{15}{16}\cdot\cdot\cdot\cdot\frac{399}{400}\)
\(=\frac{1.3}{2.2}\cdot\frac{2.4}{3.3}\cdot\frac{3.5}{4.4}\cdot\cdot\cdot\cdot\frac{19.21}{20.20}\)
\(=\frac{\left(1.2.3...19\right)\left(3.4.5...21\right)}{\left(2.3.4....20\right)\left(2.3.4....20\right)}\)
\(=\frac{1.21}{20.2}=\frac{21}{40}\)
Dễ thấy \(\frac{21}{40}>\frac{-1}{2}\)
Vậy A > -1/2
Nhầm rồi :v, làm lại
\(A=\left(\frac{1}{4}-1\right)\left(\frac{1}{9}-1\right)\left(\frac{1}{16}-1\right)....\left(\frac{1}{400}-1\right)\)
\(=\frac{-3}{4}\cdot\frac{-8}{9}\cdot\frac{-15}{16}\cdot\cdot\cdot\cdot\frac{-399}{400}\)
\(=\frac{1.\left(-3\right)}{2.2}\cdot\frac{2.\left(-4\right)}{3.3}\cdot\cdot\cdot\cdot\frac{19.\left(-21\right)}{20.20}\)
\(=\frac{\left(1.2....19\right).\left[-\left(3.4.5...21\right)\right]}{\left(2.3....20\right)\left(2.3....20\right)}\)
\(=\frac{1.\left(-21\right)}{20.2}=\frac{-21}{40}\)
Dễ thấy \(\frac{21}{40}>\frac{20}{40}\Rightarrow\frac{-21}{40}< \frac{-20}{40}=\frac{-1}{2}\)
Vậy A < -1/2
